Return Measurement Across Unequal Subperiods

Why Unequal Subperiods Need Careful Linking

When an investment account has external contributions or withdrawals, its change in value combines investment performance with changes in the amount invested. A return calculation should separate those effects. The time-weighted method does so by measuring the investment’s performance over intervals between external cash flows, linking the interval returns, and then—if needed—converting the cumulative result to an annual effective rate.

The subperiods may have different lengths. A return over three months and a return over nine months are both valid interval returns, but they do not represent equal amounts of time. The correct procedure is to link their gross return factors, not to average their percentages. Annualization is a separate step that uses the total elapsed time.

Definition: A subperiod return is the investment return over an interval whose beginning and ending valuations are measured around external cash flows. If an interval begins with value $V_{j-1}^{+}$, immediately after any flow at its start, and ends with value $V_j^{-}$, immediately before any flow at its end, its gross return factor is $G_j=V_j^{-}/V_{j-1}^{+}$, and its return is $G_j-1$. The superscripts $+$ and $-$ mean just after and just before the dated cash flow, respectively.

This calculation assumes that the external cash flow is instantaneous: it changes the account value but earns no investment return at the instant it occurs. The account is valued immediately before and after each flow, and the stated time intervals use a consistent year-fraction convention. If a cash flow is not external—for example, a dividend retained inside the account—it is part of investment performance rather than a flow to remove.

Linking and Annualizing the Returns

For $m$ consecutive subperiods, multiply their gross factors. The product gives the cumulative time-weighted growth factor over the entire measurement period. If the total elapsed time is $T$ years, the equivalent annual effective time-weighted rate is the annual rate that produces that same cumulative factor over $T$ years.

G_{\mathrm{total}}=\prod_{j=1}^{m}G_j \qquad R_{\mathrm{total}}=G_{\mathrm{total}}-1 \qquad i_{\mathrm{TW}}=G_{\mathrm{total}}^{1/T}-1

Here $R_{\mathrm{total}}$ is the cumulative return over the full period, while $i_{\mathrm{TW}}$ is an equivalent annual effective rate. If $T=1$, they are numerically equal. If $T\ne1$, they generally differ. The formula requires a positive cumulative growth factor; in ordinary account-return applications with positive valuations, each $G_j$ is positive.

Why linking works: If an interval starts with $V_{j-1}^{+}$ and ends with $V_j^{-}$, its investment growth is $V_j^{-}=G_jV_{j-1}^{+}$. Multiplying these relationships across intervals gives $V_m^{-}=G_mG_{m-1}\cdots G_1V_0^{+}$ when there are no flows between the interval boundaries. At each external flow, the post-flow value starts the next interval, so the flow itself is not mistaken for investment growth. The linked product therefore represents the growth attributable to the investment performance in each subperiod.

Mark every flow date.
Record the account value immediately before and immediately after each external contribution or withdrawal.

Form interval factors.
Divide each ending pre-flow value by the preceding beginning post-flow value. For the final interval, use the terminal value as its ending value if no flow occurs then.

Link the factors.
Multiply the subperiod gross factors to obtain the cumulative time-weighted factor.

Annualize only after linking.
Raise the cumulative factor to the power $1/T$, where $T$ is the full elapsed time in years, and subtract 1.

Worked Applications

Worked Example: A Contribution Between Unequal Subperiods

An account begins with $3,000. At time $0.25$ years its value immediately before a contribution is $3,300. The investor then contributes $700. At time $1$ year, the account is worth $3,600, with no terminal flow. Find the cumulative time-weighted return and its equivalent annual effective rate.

Method. Use the pre-flow value at $0.25$ years to end the first interval, and the post-contribution value to begin the second. The interval lengths are $0.25$ years and $0.75$ years.

The first interval factor is

G_1=\frac{3{,}300}{3{,}000}=1.10

The account value just after the contribution is $3{,}300+700=4{,}000$. Thus the second interval factor is

G_2=\frac{3{,}600}{4{,}000}=0.90

The linked factor and cumulative return are

G_{\mathrm{total}}=1.10(0.90)=0.99, \qquad R_{\mathrm{total}}=0.99-1=-0.01=-1\%

The total elapsed time is $T=1$ year, so the equivalent annual effective time-weighted rate is $0.99^{1/1}-1=-1\%$. Check: the linked factor is below 1, so a negative cumulative return is consistent. The common mistake is to treat the $700 contribution as investment growth: doing so would use the wrong opening value for the second interval.

Worked Example: A Withdrawal and a 1.5-Year Measurement Period

An account starts at $4,000. At time $0.5$ years its value immediately before a withdrawal is $4,400. The investor withdraws $400. At time $1.5$ years the account is worth $4,200. Find the cumulative time-weighted return and the equivalent annual effective rate.

The first factor is $G_1=4{,}400/4{,}000=1.10$. After the withdrawal, the account value is $4{,}400-400=4{,}000$. The second interval lasts $1.5-0.5=1$ year, and its factor is $G_2=4{,}200/4{,}000=1.05$.

G_{\mathrm{total}}=1.10(1.05)=1.155, \qquad R_{\mathrm{total}}=1.155-1=0.155=15.5\%

The elapsed time is $T=1.5$ years. Therefore,

i_{\mathrm{TW}}=1.155^{1/1.5}-1 =1.155^{2/3}-1 \approx 0.1008

The equivalent annual effective time-weighted rate is approximately 10.08%, rounded to two decimal places. Check: compounding an annual rate of about 10.08% for 1.5 years gives a factor of about 1.155. The withdrawal is not a negative investment return: it reduces the post-flow opening value for the second interval.

Worked Example: Time-Weighted Versus Dollar-Weighted Return

An account begins with $1,000. At time $0.5$ years its value immediately before a $500 contribution is $1,100. At time $1.5$ years its ending value is $1,650. Find the annualized time-weighted rate, then set up and solve for the dollar-weighted annual effective rate.

Time-weighted method. The first interval factor is $1{,}100/1{,}000=1.10$. The post-contribution value is $1{,}100+500=1{,}600$, so the second interval factor is $1{,}650/1{,}600=1.03125$. Linking gives

G_{\mathrm{total}}=1.10(1.03125)=1.134375

The period lasts $T=1.5$ years. Hence

i_{\mathrm{TW}}=1.134375^{1/1.5}-1 =1.134375^{2/3}-1 \approx 0.0877

The annualized time-weighted rate is approximately 8.77%, rounded to two decimal places.

Dollar-weighted method. The investor’s $1,000 initial investment accumulates for 1.5 years, while the $500 contribution accumulates for 1 year. If $i_{\mathrm{DW}}$ is the annual effective dollar-weighted rate, the ending value must satisfy

1{,}000(1+i_{\mathrm{DW}})^{1.5} +500(1+i_{\mathrm{DW}})=1{,}650

Solving this equation numerically gives $i_{\mathrm{DW}}\approx0.0740$, or approximately 7.40%, rounded to two decimal places. As a check, substituting $i=0.0740$ gives approximately $1{,}000(1.0740)^{1.5}+500(1.0740)\approx1{,}650$. The two rates differ because the dollar-weighted method reflects the timing and size of the investor’s contribution; the time-weighted method links interval investment performance without weighting it by the dollars contributed.

Common Traps and Practical Interpretation

A common error is to average the interval return percentages, especially when the intervals have unequal lengths. In the first worked example, the two interval returns are $10\%$ and $-10\%$. Their arithmetic average is $0\%$, but the linked cumulative return is $-1\%$, because $1.10(0.90)=0.99$, not 1. This discrepancy arises because returns compound multiplicatively. Unequal durations make a simple average even less informative: it ignores both compounding and the time represented by each interval.

Another trap is annualizing each interval and then averaging those annual rates. The annual equivalent for an interval factor $G_j$ over $t_j$ years is $G_j^{1/t_j}-1$. Those separate annual rates do not generally average to the annualized rate for the full period. The reliable calculation is to multiply the original interval factors and apply the exponent $1/T$ once, using the full elapsed time.

Measure	How it is formed	What it reflects
Cumulative time-weighted return	Product of interval factors, minus 1	Linked investment performance over the full period
Annualized time-weighted rate	Cumulative factor raised to $1/T$, minus 1	Equivalent annual effective rate for the linked performance
Dollar-weighted rate	Rate that equates dated investor cash flows and ending value	Both investment performance and the timing and size of cash flows

The time-weighted measure is useful when comparing investment performance across accounts or managers whose external cash flows differ. The dollar-weighted measure is useful when the question concerns the investor’s own experience, since a large contribution made before a strong or weak interval affects the investor’s result more than a small contribution does. Neither measure is universally preferable; the question being asked determines which one is relevant. This treatment is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Key takeaway: At each flow date, separate the pre-flow ending value from the post-flow starting value. Link the resulting gross factors, then annualize the linked factor using the total elapsed time. Do not confuse this time-weighted result with a dollar-weighted rate based on the dates and amounts of investor cash flows.

Check Your Understanding

For each account, use values immediately before and after each stated external flow. Round annualized rates to two decimal places.

An account starts at $2,500. At $0.4$ years its pre-contribution value is $2,650, then $350 is contributed. At $1.4$ years its value is $3,300. Find the cumulative time-weighted return and annualized rate. (Answer: cumulative return $16.60\%$; annualized rate approximately $11.60\%$.)
An account starts at $8,000. At $0.25$ years its pre-withdrawal value is $8,400, then $400 is withdrawn. At $1.25$ years it is worth $7,600. Find the cumulative return and annualized rate. (Answer: cumulative return $-0.25\%$; annualized rate approximately $-0.20\%$.)
An account starts at $5,000. At $0.3$ years its value before a contribution is $5,200, then $800 is contributed. At $1$ year its value before a withdrawal is $6,300, then $300 is withdrawn. At $2$ years its value is $6,600. Find the cumulative return and annualized rate. (Answer: cumulative return $20.12\%$; annualized rate approximately $9.60\%$.)
An account starts at $1,000. At year 1 its value before a $900 contribution is $1,100. At year 2 it is worth $1,800. Find the annualized time-weighted rate and the dollar-weighted rate. (Answer: time-weighted rate $-0.50\%$; dollar-weighted rate approximately $-3.49\%$.)
An account starts at $10,000. At $0.25$ years its pre-contribution value is $10,300, then $700 is contributed. At $1.25$ years its pre-withdrawal value is $12,100, then $1,100 is withdrawn. At year 2 its value is $11,550. Find its cumulative return and annualized time-weighted rate. (Answer: cumulative return $18.965\%$; annualized rate approximately $9.07\%$.)

What You'll Learn